Solving Intricate Quadratic Equations and Verifying Solutions

Quadratic equations can present a variety of challenges due to their inherent complexity. This article explores the solution process for a specific quadratic equation and highlights the potential pitfalls of extraneous solutions. We will follow a step-by-step approach, showcasing the mathematical techniques and the importance of verification in ensuring the validity of our solutions.

Introduction to the Problem

The given equation is: [ sqrt{4 - x} - sqrt{6x} sqrt{14 - 2x} ]

Step-by-Step Solution

First, we will isolate one of the square root terms. By rearranging the equation, we get:

[ sqrt{4 - x} sqrt{6x} sqrt{14 - 2x} ]

To eliminate the square roots, we square both sides of the equation:

[ 4 - x (sqrt{6x} sqrt{14 - 2x})^2 ]

Expanding the right-hand side, we get:

[ 4 - x 6x 14 - 2x 2sqrt{6x(14 - 2x)} ]

Simplifying this, we obtain:

[ 4 - x 20 - 3x 2sqrt{6x(14 - 2x)} ]

Rearranging the equation to isolate the square root term yields:

[ 2sqrt{6x(14 - 2x)} 4 - x - 20 3x ]

Simplifying further, we get:

[ 2sqrt{6x(14 - 2x)} -24 2x ]

Dividing both sides by 2:

[ sqrt{6x(14 - 2x)} -12 x ]

Next, we square both sides again to eliminate the square root:

[ 6x(14 - 2x) (-12 x)^2 ]

Expanding both sides, we get:

[ 84x - 12x^2 144 - 24x x^2 ]

Combining like terms:

[ -13x^2 108x - 144 0 ]

Multiplying the entire equation by -1:

[ 13x^2 - 108x 144 0 ]

The quadratic equation can be factored as:

[ (x - 6)(13x - 24) 0 ]

Thus, the solutions are:

[ x 6 quad text{and} quad x frac{24}{13} ]

We need to verify each potential solution in the original equation to ensure they are valid.

[ sqrt{4 - 6} - sqrt{6 cdot 6} sqrt{14 - 2 cdot 6} ]

This results in:

[ sqrt{-2} - 6 sqrt{2} ]

The left side involves an imaginary number, so ( x 6 ) is not a valid solution.

[ sqrt{4 - frac{24}{13}} - sqrt{6 cdot frac{24}{13}} sqrt{14 - 2 cdot frac{24}{13}} ]

After simplifying:

[ sqrt{frac{20}{13}} - sqrt{frac{144}{13}} sqrt{frac{302}{13}} ]

This does not simplify to a true statement, so ( x frac{24}{13} ) is also not a valid solution.

After thorough verification, we find that:

[ boxed{x -5} ] is the only valid solution.